S = 1 + 2 + 2 ^ 2 + 2 ^ 3 + ... + 2 ^ 9

2S = 2 + 2 ^ 2 + 2 ^ 3 + 2 ^ 4 + .... + 2 ^ 10

2S - S = ( 2 + 2 ^ 2 + 2 ^ 3 + 2 ^ 4 + .... + 2 ^ 10 )

           -  ( 1 + 2 + 2 ^ 2 + 2 ^ 3 + ... + 2 ^ 9 )

S        = 2 ^ 10 - 1

S       = 2 ^ 8 . 2 ^ 2 - 1

S       = 2 ^ 8 . 4 - 1

S       < 2 ^ 8 . 1 < 5 . 2 ^ 8

Vậy S < 5 . 2 ^ 8

moi nguoi oi help me

\(S=1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{45}\)

\(\frac{S}{2}=\frac{1}{2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{9\cdot10}\)

\(\frac{S}{2}=\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{10}\)

\(\frac{S}{2}=1-\frac{1}{10}\)

\(\frac{S}{2}=\frac{9}{10}\)

\(S=\frac{9}{5}

có: 2/3=1-1/3

2/15=1/3-1/5

...

=> <1

S>5 . 2⁹⁸

A = 1 + 2 + 2² + 2³ + .... +2⁹⁸

2A = 2(1 + 2 + 2² + 2³ +....+2⁹⁸)

2A = 2 + 2² + 2³ + 2⁴ +....+2⁹⁹

2A - A = (2 + 2² + 2³ + 2⁴ +.....+ 2⁹⁹) - (1 + 2 + 2² + 2³ +.....+2⁹⁸)

A = ( 2 - 2 ) +  ( 2² - 2² ) + (2³ - 2³).....+(2⁹⁸ - 2⁹⁸) + 2⁹⁹ - 1

A = 0 + 0 + 0 +.....+0  + 2⁹⁹ - 1

A = 2⁹⁹ - 1

So sánh : 

2⁹⁹ - 1 và 5 . 2⁹⁸

2⁹⁹ - 1 < 5 . 2⁹⁸

( Biết hay sai thì chịu nhe hehe phần so sánh tui 

Ta có: \(S=1+3+3^2+...+3^{20}\)

\(\Rightarrow3S=3+3^2+3^3+...+3^{21}\)

\(\Rightarrow3S-S=\left(3+3^2+3^3+...+3^{21}\right)-\left(1+3+3^2+...+3^{20}\right)\)

\(\Rightarrow2S=3^{21}-1\)

\(\Rightarrow S=\left(3^{21}-1\right).\frac{1}{2}\)

\(\Rightarrow S=3^{21}.\frac{1}{2}-\frac{1}{2}\)

Vì \(3^{21}.\frac{1}{2}-\frac{1}{2}< 3^{21}.\frac{1}{2}\) nên \(A< \frac{1}{2}.3^{21}\)

Vậy \(A< \frac{1}{2}.3^{21}\)

\(S=1+2+2^2+2^3+...+2^9\)

\(2S=2+2^2+2^3+2^4+...+2^{10}\)

\(2S-S=\left(2+2^2+2^3+...+2^{10}\right)-\left(1+2+2^2+...+2^9\right)\)

\(2S-S=2+2^2+2^3+...+2^{10}-1-2-2^2-...-2^9\)

\(S=2^{10}-1\)

\(P=4.\frac{5}{4}.2^8\)

\(P=2^2.2^8.\frac{5}{4}=2^{10}.\frac{5}{4}\)

\(\Rightarrow S< P\)

S < P nhe

Đảm bảo 100% là đúng

S = 1 + 2 + 2^2 + 2^3 + ... + 2 ^ 9 . 

S = 2^10 - 1 .